How To Find Mean Of Two Sets Of Data

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Are you lot taking the Saturday or Act and desire to brand sure you know how to work with information sets? Or maybe you lot're looking to refresh your retentiveness for a high schoolhouse or college math class. Whatsoever the instance, information technology'due south important you know how to detect the mean of a data set.

We'll explain what the mean is used for in math, how to calculate the mean, and what problems nearly the hateful can look similar.

What Is a Mean and What Is It Used For?

The hateful, or arithmetic hateful, is the average value of a set of numbers. More specifically, it'southward the measure of a "central" or typical tendency in a given set of data.

Mean— often simply chosen the "average"—is a term used in statistics and data assay. In addition, it'southward not unusual to hear the words "mean" or "boilerplate" used with the terms "manner," "median," and "range," which are other methods of calculating the patterns and common values in data sets.

Briefly, here are the definitions of these terms:

Mode— the value that appears most frequently in a data ready
Median— the center value of a data ready (when arranged from everyman value to highest)
Range— the deviation between the highest and smallest values in a data gear up

And then what is the purpose of the mean exactly? If you have a data set with a wide range of numbers, knowing the mean can give you a general sense of how these numbers could essentially exist put together into a single representative value.

For example, if you're a high school student getting prepare to accept the Sat, you might be interested to know the current mean Saturday score. Knowing the mean score gives you a rough thought of how nigh students taking the SAT tend to score on it.

How to Find the Mean: Overview

To detect the arithmetic mean of a information set, all yous need to do is add together upwardly all the numbers in the information gear up and so carve up the sum by the full number of values.

Allow'southward expect at an case. Say you're given the following set of data:

$$half dozen, 10, 3, 27, xix, 2, 5, 14$$

To find the mean, y'all'll offset demand to add upward all the values in the data prepare like this:

$$6 + x + 3 + 27 + 19 + 2 + 5 + 14$$

Annotation that you don't need to rearrange the values here (though you may if yous wish to) and can only add together them in the order in which they've been presented to yous.

Adjacent, write down the sum of all the values:

$$6 + 10 + three + 27 + 19 + 2 + five + 14 = \bo86$$

The final step is to take this sum (86) and separate information technology past the number of values in the data set. Because there are eight different values (6, x, iii, 27, 19, ii, 5, fourteen), nosotros'll be dividing 86 by 8:

$$86 / viii = 10.75$$

The mean, or average, for this fix of data is x.75.

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How to Calculate a Mean: Practice Questions

Now that y'all know how to find the average—in other words,how to calculate the mean of a given set of information—it'due south time to examination what y'all've learned. In this section, nosotros'll give y'all four math questions that involve finding or using the mean.

The beginning two questions are our own, whereas the second two are official SAT/ACT questions; as such, these two will require a lilliputian fleck more thought.

Scroll past the questions for the answers and respond explanations.

Practice Question one

Observe the mean of the following set of numbers: 5, 26, 9, 14, 49, 31, 109, v.

Practise Question two

You are given the following list of numbers: 4, 4, 2, 11, half-dozen, $X$, 1, 3, 2. The arithmetic hateful is four. What is the value of $10$?

Practice Question 3

The list of numbers 41, 35, xxx, $X$,$Y$, fifteen has a median of 25. The fashion of the list of numbers is 15. To the nearest whole number, what is the mean of the list?

twenty
25
26
27
thirty

Source: 2022-xix Official ACT Practice Test

Practice Question 4

At a primate reserve, the mean age of all the male primates is xv years, and the mean age of all female primates is xix years. Which of the following must be true almost the mean age $thou$ of the combined grouping of male person and female person primates at the primate reserve?

$m = 17$
$one thousand > 17$
$m < 17$
$fifteen < thou < xix$

Source: The College Lath

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How to Observe the Boilerplate: Answers + Explanations

One time y'all've tried out the four exercise questions to a higher place, it's time to compare your answers and see whether yous sympathize non just how to find the mean of information but besides how to use what y'all know almost the mean to more than effectively approach any math questions that deal with averages.

Hither are the answers to the 4 do questions in a higher place:

Practice Question 1: 31
Practice Question 2: 3
Practice Question iii: C. 26
Do Question four: D. $fifteen < k < 19$

Keep reading to see the reply explanation for each question.

Practice Question 1 Reply Explanation

Notice the mean of the post-obit gear up of numbers: 5, 26, 9, 14, 49, 31, 109, 5.

This is a straightforward question that simply asks you to calculate the arithmetics mean of a given data ready.

First, add upwardly all the numbers in the data set (remember that you don't need to suit them in guild from lowest to highest—just do this if you're trying to observe the median):

$$5 + 26 + 9 + 14 + 49 + 31 + 109 + five = \bo248$$

Side by side, take this sum and divide it by the number of values in the data set. Here, there are eight total values, and then we'll divide 248 by eight:

$$248 / 8 = 31$$

The mean and right answer is 31.

Exercise Question 2 Respond Explanation

You are given the following list of numbers: 4, 4, 2, 11, 6, $X$, 1, 3, 2. The arithmetic hateful is 4. What is the value of $Ten$?

For this question, you lot're substantially working astern: you already know the hateful and now must use this knowledge to aid you solve for the missing value, $X$, in the information set.

Recall that to find the hateful, y'all add up all the numbers in a set and and so separate the sum by the total number of values.

Since we know the mean is 4, we'll first by multiplying four by the number of values (in that location are nine separate numbers hither, including $X$):

$$4 * 9 = 36$$

This gives u.s.a. the sum of the data set (36). Now, the question becomes an algebra trouble, in which all we need to do is simplify and solve for $10$:

$$iv + 4 + two + xi + 6 + X + 1 + 3 + 2 = 36$$

$$33 + X = 36$$

$$X = 3$$

The correct answer is 3.

body_math_practice Practice makes perfect!

Practice Question three Answer Explanation

The list of numbers 41, 35, 30, $X$, $Y$, 15 has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the hateful of the listing?

xx
25
26
27
thirty

This catchy-looking math problem comes from an official ACT practice test, then you lot can expect it to be a little less direct than your typical arithmetic mean problem.

Here, we're given a data set with two unknown values:

41, 35, xxx, $X$, $Y$, 15

We're as well given two critical pieces of data:

The manner is 15
The median is 25

To solve for the mean of this information set, we volition need to employ all the information we've been given and volition also need to know what the mode and median are.

Equally a reminder, the mode is the value that appears most frequently in a data set, while the median is the heart value in a data set (when all values have been arranged from lowest to highest).

Since the mode is fifteen, this must mean that the value xv appears at least twice in the data prepare (in other words, more than times than any other value appears). Every bit a event, we tin say replace either $X$ or $Y$ with 15:

41, 35, 30, $X$,15,15

We're also told that the median is 25. To notice the median, yous must first rearrange the information set in order from lowest value to highest value.

Since the median is more than than 15 but less than 30, we should put $X$ between these two values. Here's what we get when we rearrange our values from lowest to highest:

15, fifteen, $X$, 30, 35, 41

At that place are six values in total, (including $X$) significant that the median will be the number exactly halfway betwixt the third and fourth values in the data set.In short,25 (the median) must come halfway betwixt $X$ and xxx.

This means that $X$ must equal twenty, since that would put it 5 away from xx and 5 away from xxx (or halfway between the two values).

We now accept a complete data gear up with no unknown values:

15,15, twenty, 30, 35, 41

All nosotros accept to do at present is apply these values to solve for the mean. Start past calculation them all upwardly:

xv+fifteen+20+xxx+35+41=156

Finally, separate the sum by the number of values in the data set (that'south 6):

156/half-dozen=26

The correct answer is C. 26.

Practice Question 4 Answer Caption

At a primate reserve, the mean age of all the male primates is 15 years, and the hateful age of all female primates is 19 years. Which of the following must exist true near the mean historic period $k$ of the combined group of male and female primates at the primate reserve?

$m = 17$
$one thousand > 17$
$k < 17$
$fifteen < yard < 19$

This practice problem is an official Sabbatum Math exercise question from the College Board website.

For this math question, y'all're non expected to solve for the hateful but must instead apply what you know about two means to explain what the mean of the larger group could exist. Specifically, we're being asked how we tin can apply these two means to express, in algebraic terms, the mean age ($\bi one thousand$) forboth male person and female primates.

Hither's what we know: first, the mean age of all male primates is fifteen years. Secondly, the mean age of all female primates is 19 years. This means that, in general, the female primates are older than the male primates.

Since the mean age for male primates (xv) is lower than that for female primates (19), we know that the mean age for both groups cannot logically exceed 19 years.

Similarly, because the hateful age for female primates is greater than that for male primates, nosotros know that the mean age for both cannot logically autumn below 15 years.

Nosotros are therefore left with the understanding that the mean historic period for the male person and female person primates together must be greater than fifteen years (the mean age of the males) but likewise less than xix years (the mean historic period of the females).

This rationale can be written as the following inequality:

$$15 < chiliad < nineteen$$

The correct answer is D. 15 < $\bi m$ < 19.

What's Side by side?

To acquire fifty-fifty more about information sets,look at our guide to the all-time strategies for mean, median, and manner on Sat Math.

Taking the Sat or ACT soon? So you'll definitely want to know what kind of math you're going to exist tested on. Check out our in-depth guides to the SAT Math section and the Deed Math department to get started.

What are the most important math formulas to know for the Saturday and Human activity?Go an overview of the 28 critical Saturday formulas and the 31 critical Human activity formulas you should know.

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About the Writer

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor'southward caste from the University of Southern California. From 2022 to 2022, she taught English in Nihon via the JET Programme. She is passionate near education, writing, and travel.

Source: https://blog.prepscholar.com/how-to-find-the-mean